Schur-convexity of 2nd order, certain subclass of multivariate arrangement increasing functions with applications in statistics

Abstract

It is shown that starting with a certain meaningful problem of the type “ranking of populations”, a need arises to employ functions which we call “Schur-convex of 2nd order with respect to two variables”. These functions L(v1,v2,v3,…,vn) are symmetric, and they are characterized in essence by the relation Lv12″−2Lv1v2″+Lv22″≥0. It is shown that this subclass of Schur-convex functions is closely related to a certain subclass of multivariate arrangement increasing functions introduced by Boland and Proschan [P.J. Boland, F. Proschan, Multivariate arrangement increasing functions with applications in probability and statistics, J. Multivariate Anal. (1988) 25 286–298]. This relation allows us to solve a series of statistical problems concerning maximization of the goal function with respect to the risk criterion on the set of permutations of the function’s arguments.For families of distributions with location parameter, it is shown that the ranking of parameters by sample means is expedient in the case of families having no sufficient statistic. Furthermore, mean losses (the risk) arising because of making wrong decisions decreases with growth of the size of the sample if densities are symmetric and log- concave. Examples of applications of these results are given.